Ladder Bethe-Salpeter Equation Research Articles

Resonance enhancement of the electromagnetic interaction between two charged particles in the bound state in the continuum

In bound states in the continuum (BIC) the mass of a composite particle is greater than the total mass of its constituents. The BIC state is investigated in a system of two charged particles [Formula: see text] and [Formula: see text] with different masses ([Formula: see text]), using the ladder Bethe–Salpeter equation. We demonstrate that there are two momentum space regions for the positive binding energy, in which the electromagnetic interaction between the particles is strongly enhanced, and the effective coupling constant turns out to be equal to [Formula: see text], where [Formula: see text] is the fine structure constant. The interaction resonance confines the constituents in the BIC state with the positive binding energy, which is of the order of the mass of the lighter particle from this pair.

Electronic structure of chromium trihalides beyond density functional theory

We explore the electronic band structure of free standing monolayers of chromium trihalides, CrX\textsubscript{3}{, X= Cl, Br, I}, within an advanced \emph{ab-initio} theoretical approach based in the use of Green's function functionals. We compare the local density approximation with the quasi-particle self-consistent \emph{GW} approximation (QS\emph{GW}) and its self-consistent extension (QS$G\widehat{W}$) by solving the particle-hole ladder Bethe-Salpeter equations to improve the effective interaction \emph{W}. We show that at all levels of theory, the valence band consistently changes shape in the sequence Cl{\textrightarrow}Br{\textrightarrow}I, and the valence band maximum shifts from the M point to the $\Gamma$ point. However, the details of the transition, the one-particle bandgap, and the eigenfunctions change considerably going up the ladder to higher levels of theory. The eigenfunctions become more directional, and at the M point there is a strong anisotropy in the effective mass. Also the dynamic and momentum dependent self energy shows that QS$G\widehat{W}$ adds to the localization of the systems in comparison to the QS\emph{GW} thereby leading to a narrower band and reduced amount of halogens in the valence band manifold.

Solving the Bethe-Salpeter equation in Minkowski space for a fermion-scalar system

The ladder Bethe-Salpeter equation of a bound $(1/2{)}^{+}$ system, composed by a fermion and a scalar boson, is solved in Minkowski space, for the first time. The formal tools are the same already successfully adopted for two-scalar and two-fermion systems, namely the Nakanishi integral representation of the Bethe-Salpeter amplitude and the light-front projection of the fulfilled equation. Numerical results are presented and discussed for two interaction kernels: (i) a massive scalar exchange and (ii) a massive vector exchange, illustrating both the correlation between binding energies and the interaction coupling constants, as well as the valence content of the interacting state, through the valence probabilities and the light-front momentum distributions. In the case of the scalar exchange, an interesting side effect, to be ascribed to the repulsion generated by the small components of the Dirac spinor, is pointed out, while for the vector exchange the manifestation of the helicity conservation opens new interesting questions to be addressed within a fully nonperturbative framework, as well as the onset of a scale-invariant regime.

Quark-antiquark bound states and QCD running coupling within infrared confinement

Bound states of spin-half and spin-one particles are studied within a relativistic quantum-field model based on the infrared confinement. The spectra of quark-antiquark and two-gluon states are determined by master equations similar to the ladder Bethe-Salpeter equations. Masses of light, intermediate and heavy mesons are estimated in a wide range of scale (from hundreds MeV up to 10 GeV). A new, independent and specific infrared-finite behavior of QCD effective coupling is revealed in the low-energy domain.

Hadron Spectrum and the Infrared Behavior of QCD Coupling

A relativistic quantum-field model based on analytical confinement is developed to build stable bound states of spin-half and spin-one particles. The spectra of two-quark ground states are defined by using the ladder Bethe-Salpeter equation. An analytic expression for the running coupling in QCD is obtained that can be extended to the low energy region. By comparing the obtained result with recent experimental data on meson masses we obtain a new, independent and specific infrared-finite behavior of QCD coupling below energy scale ∼ 1 GeV. Particularly, an infrared-fixed point is extracted at αS(0) ≃ 0.757. We also calculate masses of some conventional mesons in the range of 0.8 – 9.5 GeV.

QCD running coupling in low-energy region

We estimate the QCD effective charge $\alpha_s$ in the low-energy region by exploiting the conventional meson spectrum within a relativistic quantum-field model based on analytic confinement. The ladder Bethe-Salpeter equation is solved for the masses of two-quark bound states. We found a new, independent and specific infrared-finite behavior of QCD coupling below energy scale 1 GeV. Particularly, an infrared-fixed point is extracted at $\alpha_s(0)\simeq 0.757$ for confinement scale $\Lambda=345$ MeV. As an application, we estimate masses of some intermediate and heavy mesons and obtain results in reasonable agreement with recent experimental data.

Two-particle Bound States: Mesons and Glueballs

A relativistic quantum-field model based on analytic confinement is considered to study the two- quark and two-gluon bound states. For the spectra of two-particle bound states we solve the ladder Bethe-Salpeter equation. We provide a new, independent and analytic estimate of the lowest glueball mass and found it at 1660 MeV. The conventional mesons and the weak decay constants are described to extend the consideration. By using a few parameters (the quark masses, the coupling constant and the confinement scale) we obtain numerical results which are in reasonable agreement with experimental evidence in the wide range of energy scale from 140MeV up to 9 GeV. The model can serve a reasonable framework to describe simultaneously di erent sectors in low-energy particle physics.

Manifestation of Three-Body Forces in Three-Body Bethe–Salpeter and Light-Front Equations

Bethe-Salpeter and light-front bound state equations for three scalar particles interacting by scalar exchange-bosons are solved in ladder truncation. In contrast to two-body systems, the three-body binding energies obtained in these two approaches differ significantly from each other: the ladder kernel in light-front dynamics underbinds by approximately a factor of two compared to the ladder Bethe-Salpeter equation. By taking into account three-body forces in the light-front approach, generated by two exchange-bosons in flight, we find that most of this difference disappears; for small exchange masses, the obtained binding energies coincide with each other.

Glueballs and mesons: The ground states

We provide a new, independent, and analytic estimate of the lowest glueball mass, and we found it at 1661 MeV within a relativistic quantum-field model based on analytic confinement. The conventional mesons and the weak decay constants are described to extend the consideration. For the spectra of two-gluon and two-quark bound states we solve the ladder Bethe-Salpeter equation. By using a minimal set of model parameters (the quark masses, the coupling constant, and the confinement scale) we obtain numerical results which are in reasonable agreement with experimental evidence in the wide range of energy scale. The model serves a reasonable framework to describe simultaneously different sectors in low-energy particle physics.

Structure and decay constant of the ρ meson within the Bethe-Salpeter equation

In this article, we study the structure of the \ensuremath{\rho} meson in the framework of the coupled rainbow Schwinger-Dyson equation and ladder Bethe-Salpeter equation with a confining effective potential. The $u$ and $d$ quark propagators get significantly modified, the mass poles are absent in the timelike region, which implements confinement naturally. The Bethe-Salpeter amplitudes of the \ensuremath{\rho} meson center around zero momentum and extend to the energy scale about ${q}^{2}=1\phantom{\rule{0.3em}{0ex}}{\mathrm{GeV}}^{2}$, which happens to be the energy scale of chiral symmetry breaking, strong interactions in the infrared region result in bound state. The numerical results of the mass and decay constant of the \ensuremath{\rho} meson are in agreement with the experimental data.

Decay constants of the pseudoscalar charmonium and bottomonium

In this Letter, we investigate the structures of the pseudoscalar charmonium and bottomonium in the framework of the coupled rainbow Schwinger–Dyson equation and ladder Bethe–Salpeter equation with the confining effective potential (infrared modified flat bottom potential). As the current masses are very large, the dressing or renormalization for the c and b quarks are tender, however, mass poles in the timelike region are absent. The Euclidean time Fourier transformed quark propagator has no mass poles in the timelike region which naturally implements confinement. The Bethe–Salpeter wavefunctions for those mesons have the same type (Gaussian type) momentum dependence and center around zero momentum with spatial extension to about q2=1 GeV2 which happen to be the energy scale for chiral symmetry breaking, the strong interactions in the infrared region result in bound states. The decay constants for those pseudoscalar heavy quarkonia are compatible with the values of experimental extractions and theoretical calculations.

Decay constants of the pseudoscalar mesons in the framework of the coupled Schwinger–Dyson equation and Bethe–Salpeter equation

In this article, we investigate the structures of the pseudoscalar mesons ( π, K, D, D s , B and B s ) in the framework of the coupled rainbow Schwinger–Dyson equation and ladder Bethe–Salpeter equation with the confining effective potential (infrared modified flat bottom potential). The Schwinger–Dyson functions for the u, d and s quarks are greatly renormalized at small momentum region and the curves are steep at about q 2 = 1 GeV 2 which indicates an explicitly dynamical symmetry breaking. The Euclidean time Fourier transformed quark propagators have no mass poles in the time-like region which naturally implements confinement. As for the c and b quarks, the current masses are very large, the renormalization are more tender, however, mass poles in the time-like region are also absent. The Bethe–Salpeter wavefunctions for those mesons have the same type (Gaussian type) momentum dependence and center around small momentum which indicate that the bound states exist in the infrared region. The decay constants for those pseudoscalar mesons are compatible with the values of experimental extractions and theoretical calculations, such as lattice simulations and QCD sum rules.

Decay constants of the pion and B mesons with the Bethe–Salpeter equation

In this Letter, we investigate the under-structures of the π and B mesons in the framework of the Bethe–Salpeter equation with the confining effective potential (infrared modified flat bottom potential). In bare quark–gluon vertex approximation, we obtain the algebraic expressions for the solutions of the coupled rainbow Schwinger–Dyson equation and ladder Bethe–Salpeter equation. Firstly, we neglect the rainbow Schwinger–Dyson equation, take the bare quark propagator and solve the Bethe–Salpeter equation numerically alone. Although the bare quark propagator cannot embody dynamical chiral symmetry breaking and has a mass pole in the time-like region, it can give reasonable results for the values of decay constants fπ and fB compared with the values of experimental data and other theoretical calculations, such as lattice simulations and QCD sum rules. Secondly, we explore those mesons within the framework of the coupled rainbow Schwinger–Dyson equation and ladder Bethe–Salpeter equation. The Schwinger–Dyson functions for the u and d quarks are greatly renormalized at small momentum region and the curves are steep at about q2=1 GeV2 which indicates an explicitly dynamical symmetry breaking. The Euclidean time Fourier-transformed quark propagator has no mass poles in the time-like region which naturally implements confinement. As for the b quark, the current mass is very large, the renormalization is more tender, however, mass pole in the time-like region is also absent. The Bethe–Salpeter wavefunctions for both the π and B mesons have the same type (Gaussian type) momentum dependence as the corresponding wavefunctions with the bare quark propagator, however, the quantitative values are changed and the values for the decay constants fπ and fB are changed correspondingly.

Analysis of a quenched lattice-QCD dressed-quark propagator

Quenched lattice-QCD data on the dressed-quark Schwinger function can be correlated with dressed-gluon data via a rainbow gap equation so long as that equation's kernel possesses enhancement at infrared momenta above that exhibited by the gluon alone. The required enhancement can be ascribed to a dressing of the quark-gluon vertex. The solutions of the rainbow gap equation exhibit dynamical chiral symmetry breaking and are consistent with confinement. The gap equation and related, symmetry-preserving ladder Bethe-Salpeter equation yield estimates for chiral and physical pion observables that suggest these quantities are materially underestimated in the quenched theory: |<bar-q q>| by a factor of two and f_pi by 30%.

Study of relativistic bound states for scalar theories in the Bethe-Salpeter and Dyson-Schwinger formalism

The Bethe-Salpeter equation for Wick-Cutkosky like models is solved in dressed ladder approximation. The bare vertex truncation of the Dyson-Schwinger equations for propagators is combined with the dressed ladder Bethe-Salpeter equation for the scalar S-wave bound state amplitudes. With the help of spectral representation the results are obtained directly in Minkowski space. We give a new analytic formula for the resulting equation simplifying the numerical treatment. The bare ladder approximation of Bethe-Salpeter equation is compared with the one with dressed ladder. The elastic electromagnetic form factors is calculated within the relativistic impulse approximation.

Kl3transition form factors

The rainbow truncation of the quark Dyson-Schwinger equation is combined with the ladder Bethe-Salpeter equation for the meson bound state amplitudes and the dressed quark-W vertex in a manifestly covariant calculation of the K_{l3} transition form factors and decay width in impulse approximation. With model gluon parameters previously fixed by the chiral condensate, the pion mass and decay constant, and the kaon mass, our results for the K_{l3} form factors and the kaon semileptonic decay width are in good agreement with the experimental data.

Lower bound to the mass of a relativistic three-boson system in thelightcone

The lower bound masses of the ground-state relativisticthree-boson system in 1 + 1, 2 + 1 and 3 + 1 spacetimedimensions are obtained. We have considered a reductionof the ladder Bethe-Salpeter equation to the lightfrontin a model with renormalized two-body contactinteraction. The lower bounds are deduced with theconstraint of reality of the two-boson subsystem mass.It is verified that, in some cases, the lower boundapproaches the ground-state binding energy. Thecorresponding non-relativistic limits are also verified.

π,K+,andK0electromagnetic form factors

The rainbow truncation of the quark Dyson-Schwinger equation is combined with the ladder Bethe-Salpeter equation for the meson amplitudes and the dressed quark-photon vertex in a self-consistent Poincar\'e-invariant study of the pion and kaon electromagnetic form factors in impulse approximation. We demonstrate explicitly that the current is conserved in this approach and that the obtained results are independent of the momentum partitioning in the Bethe-Salpeter amplitudes. With model gluon parameters previously fixed by the condensate, the pion mass and decay constant, and the kaon mass, the charge radii and spacelike form factors are found to be in good agreement with the experimental data.

A comparison between relativistic and semi-relativistic treatment in the diquark–quark model

In the diquark–quark model of the nucleon including scalar and axialvector diquarks we compare solutions of the ladder Bethe–Salpeter equation in the instantaneous Salpeter approximation and in the fully covariant (i.e. four-dimensional) treatment. We obtain that the binding energy is severely underestimated in the Salpeter approximation. For the electromagnetic form factors of the nucleon we find that in both approaches the overall shapes of the respective form factors are reasonably similar up to Q 2≈0.4 GeV 2. However, the magnetic moments differ substantially as well as results for the pion–nucleon and the axial coupling of the nucleon.

Quark-photon vertex and the pion charge radius

The rainbow truncation of the quark Dyson-Schwinger equation is combined with the ladder Bethe-Salpeter equation for the dressed quark-photon vertex to study the low-momentum behavior of the pion electromagnetic form factor. With model gluon parameters previously fixed by the pion mass and decay constant, the pion charge radius ${r}_{\ensuremath{\pi}}$ is found to be in good agreement with the data. In this work, about half of ${r}_{\ensuremath{\pi}}^{2}$ is seen to be attributable to the presence of the $\ensuremath{\rho}$ pole in the solution of the ladder Bethe-Salpeter equation.